Oct 16, 2016 - Black-Schaffer.35,36. A Majorana .... âhatâ for 2Ã2 matrices),. ËHS(k) = ( ..... 43 S. Nadj-Perge, I. K. Drozdov, J. Li, H. Chen, S. Jeon, J. Seo, A.
Unconventional pairing in three-dimensional topological insulators with warped surface state A. S. Vasenko,1 A. A. Golubov,2, 3 V. M. Silkin,4, 5, 6 and E. V. Chulkov4, 5, 7 1 National
Research University Higher School of Economics, 101000 Moscow, Russia of Science and Technology and MESA+ Institute for Nanotechnology, University of Twente, 7500 AE Enschede, The Netherlands 3 Moscow Institute of Physics and Technology, Dolgoprudny, 141700 Moscow, Russia 4 Donostia International Physics Center (DIPC), Paseo Manuel de Lardizabal 4, San Sebasti´an/Donostia, 20018 Basque Country, Spain 5 Departamento de F´ıisica de Materiales, Facultad de Ciencias Qu´ıimicas, Universidad del Pais Vasco/Euskal Herriko Unibertsitatea, Apdo. 1072, San Sebasti´an/Donostia, 20080 Basque Country, Spain 6 IKERBASQUE, Basque Foundation for Science, 48011 Bilbao, Spain 7 Tomsk State University, 634050 Tomsk, Russia (Dated: October 18, 2016)
arXiv:1606.00905v3 [cond-mat.supr-con] 16 Oct 2016
2 Faculty
We study the effect of the Fermi surface anisotropy (hexagonal warping) on the superconducting pair potential, induced in a three-dimensional topological insulator (TI) by proximity with a superconductor (S) in the presence of a magnetic moment, when a superconductor/ ferromagnetic insulator (S/FI) hybrid structure is formed on the TI surface. In the previous studies similar problem was treated with a simplified Hamiltonian, describing an isotropic Dirac cone dispersion. This approximation is only valid near the Dirac point. However, in topological insulators the chemical potential often lies well above this point, where the Dirac cone is strongly anisotropic and its constant energy contour has a snowflake shape. Taking this shape into account we show that in the structure under consideration a very exotic pair potential is induced in the topological insulator surface. It has consequences on the Majorana bound state and may cause the spontaneous supercurrent flowing along the S/FI boundary of the structure. PACS numbers: 73.20.-r, 74.45.+c, 74.78.Fk, 03.65.Vf
I.
INTRODUCTION
Three-dimensional topological insulators (TI) represent a recently discovered new state of matter.1–5 Their hallmark is the formation of conducting surface states with the Dirac dispersion relation (similar to graphene), whereas the bulk states are gapped. Recently a lot of interest has been attracted to the physics of hybrid structures involving topological insulators and other materials, for example new electronic states have been predicted to appear in TI contacts to ferromagnets and superconductors, such as a magnetic monopole6 and a Majorana fermion (MF).7–11 A MF has been a primary focus of many extensive studies, since it was proposed as a building block of a topological qubit, that is robust against local decoherence.12 When the topological insulator is placed in the electrical contact with the superconductor (S), the superconducting pair correlations penetrate into the topological state due to the proximity effect. Its key mechanism is the Andreev reflection process, which provides the possibility for converting single electron states from a topological insulator to Cooper pairs in the superconducting condensate. Recently the S/TI proximity effect was studied by many authors and the formation of an exotic induced pair potential with the so called oddfrequency pairing in the TI surface was predicted in some particular cases, for example in the presence of the external magnetic field.13–20 In accordance with the Pauli principle, it is customary to distinguish spin-singlet even-parity and spin-triplet odd-parity pairing states, where odd (even) refer to the orbital part of the
pair wave function. For example, pairing states in the s-wave and d-wave superconductors belong to the former case while pairing states in the p-wave superconductor belongs to the latter one.21 In both cases, the pair amplitude is an even function of energy (or Matsubara frequency). However, the so-called odd-frequency pairing states when the pair amplitude is an odd function of energy can also exist. Then, the spin-singlet odd-parity and the spin-triplet even-parity pairing states are allowed by the Pauli principle. The possibility of realizing the odd-frequency pairing state was first proposed by Berezinskii in the context of 3 He.22 Although the existence of odd-frequency pairing in bulk uniform systems is not fully established yet,23–29 there is a number of ways to realize it in superconducting junctions. For example the odd-frequency pairing were demonstrated in superconductor/ ferromagnetic metal hybrid structures.31 Recently, it was also shown that the odd-frequency spin-singlet odd-parity pairing is quite generally induced near the normal metal/ superconductor interface during the formation of the Andreev bound states.32–34 In S/TI heterostructures another type of odd-frequency pairing is induced in the presence of the external magnetic field, perpendicular to the topological insulator surface - the odd-frequency spin-triplet even-parity pairing state.13–18 It was shown by Asano and Tanaka that in a one-dimensional nano-wire, proximity coupled to a topological superconductor, the odd-frequency spin-triplet even-parity pairing and the Majorana fermion are two sides of a same coin.13 It was also argued that similar assumption is valid for 2D topological surface states of a three-dimensional TI in proximity with an s-wave superconductor.15–17 The proximity induced unconventional superconductivity in topologi-
2 cal insulators was also extensively studied by Balatsky and Black-Schaffer.35,36 A Majorana fermion is a topological state that is its own anti-particle, in striking contrast to any known fermion so far. Generally, in solid state physics, electronic transport can either be described in terms of electrons or in terms of holes. In order for a Majorana fermion to exist, it would have to be simultaneously half-electron and half-hole and electrically neutral. Therefore, the zero energy is a likely place to look for a Majorana fermion. Experimentalists have already reported signatures of the Majorana zero-energy mode, where zerobias conductance peaks are the main features observed in this context.37–40 Signatures of Majorana zero modes were also observed in the Josephson effect in HgTe-based junctions,41,42 in ferromagnetic atomic chains formed on a superconductor,43 and in a semiconductor Coulomb island in proximity with a superconductor.44 Impressive number of theoretical studies of electron transport in different hybrid devices containing Majorana fermions was published in recent years.45–62 Majorana fermions in superconductor/ topological insulator hybrid structures have been first predicted by Fu and Kane45 as zero energy states at the site of a vortex, induced by the magnetic field on the surface of the topological insulator in proximity with a superconductor. A Majorana fermion was also predicted to occur if the externally applied magnetic field is replaced by the magnetic moment of a nearby ferromagnet (F) or a ferromagnetic insulator (FI). In the latter case, the Majorana fermion turns out to be a one-dimensional linearly dispersing mode along the S/FI boundary, when a S/FI junction is formed on the topological insulator surface.46 The ferromagnetic insulator was taken to insure that the current is only going through the surface states. Bringing together the topological surface states, superconductivity and magnetic field, perpendicular to the TI surface, also required for the realization of the odd-frequency spin-triplet even-parity pairing state near the S/TI interface.13–17 In many aforementioned studies the theoretical predictions about the proximity effect, induced pairing potential and Majorana fermion were made on the basis of a simplified model, when the topological surface states were described with an isotropic Dirac cone. Within k · p theory a 2×2 Hamiltonian of surface states in this model to the lowest order in k reads Hˆ 0 (k) = −µ + v(kx σˆ x + ky σˆ y ) [so called Dirac-type Hamiltonian]. Here µ is a chemical potential, v is a Fermi velocity, k = (kx , ky ) denotes in-plane quasiparticle momentum, and σˆ x,y are the Pauli matrices in spin space. It should be noted that it is also possible to use a Bychkov-Rashba term63 in the Hamiltonian of a topological insulator, Hˆ 0 (k) = − µ + v(kx σˆ y − ky σˆ x ). This gives rise to a different spin-momentum locking on the Fermi surface.3,4 Nevertheless, all conclusions on the excitation spectrum are independent of whether one uses the Dirac or Bychkov-Rashba type for Hˆ 0 (k).64 However, such isotropic forms of the Hamiltonian are only valid if the chemical potential lies near the Dirac point, while in realistic topological insulators it usually lies well above this point, where the Dirac cone distortion can’t be any more neglected. To develop more realistic theoretical description of the superconducting proximity effect in the topological insula-
Bulk conduction band
Warped Dirac cone FIG. 1: (Color online) The hexagonal warped Dirac cone in threedimensional topological insulators, shown schematically. A set of constant energy contours for different chemical potential positions (shown by dashed blue lines) is presented. Near the Dirac point the constant energy contour is almost circular, evolving to hexagonal with increasing energy, and then to snowflake near the bulk conduction band.
tor surface states it is important to take into account the Dirac cone anisotropy. For example, the Fermi surface of Bi2 Te3 topological insulator observed by angle resolved photoemission spectroscopy (ARPES) is nearly a hexagon, having snowflake-like shape: it has relatively sharp tips extending along six directions and curves inward in between.65–67 Moreover, the shape of constant energy contour is energydependent, evolving from a snowflake to a hexagon and then to a circle near the Dirac point, see Fig. 1. Later same anisotropy was found in Bi2 Se3 ,68,69 Pb(Bi,Sb)2 Te4 70 and other topological insulator materials.71 Recently it was realized that the aforementioned simplified Hamiltonian can be extended to higher order terms in the momentum. Namely, Fu found an unconventional hexagonal warping term Hˆ w (k) in the surface band structure, which is the counterpart of cubic Dresselhaus spin-orbit coupling in rhombohedral structures.72 The effective Hamiltonian of surface states then reads, ˆ H(k) = − µ + v(kx σˆ y − ky σˆ x ) + Hˆ w (k),
(1)
where the hexagonal warping term
λ 3 3 + k− )σˆ z , Hˆ w (k) = (k+ 2
(2)
k± = kx ± iky , σˆ = (σˆ x , σˆ y , σˆ z ) are the Pauli matrices in spin space, and λ is the hexagonal warping strength. The Hamiltonian in Eq. (1) describes perfectly the warped Dirac cone as shown in Fig. 1. It is the consequence of the rhombohedral crystal structure symmetry, typical to three-dimensional topological insulators. Representative values for the material parameters v and λ can be inferred from ARPES (see for instance Ref. 71 and references therein). We stress here that the term in Eq. (2) is different from the trigonal warping term in graphene.72,73 It is interesting to mention that the trigonal
3 warping was predicted to cause the Majorana zero modes in single layer graphene.74 The warping term, Eq. (2), breaks the rotational symmetry of the Dirac cone. Moreover, it is an odd-parity term, since it is odd under transformation k → −k. It is then obvious that addition of this term may dramatically change the physical properties of topological insulator surface states. The consequences of warping on magnetic72,75,76 and transport properties77,78 of topological insulators have been discussed extensively in the literature. Recently the magnetic order on a topological insulator surface with hexagonal warping and proximity-induced superconductivity was also studied in Ref. 79. In this work we study the interplay between topological order and superconducting correlations performing a symmetry analysis of the induced pair potential based on the anomalous Green function and analyze the conditions of possible experimental realization of the Majorana mode in a hybrid structure, where a S/FI junction is formed on the topological surface as shown in Fig. 2(a).46 We take into account the hexagonal warping effect, which was never considered previously in connection with induced superconducting pairing and a MF realization, for instance in the original Ref. 46. We show the formation of a very exotic pair potential with the existence of all possible pairings, allowed by the Pauli principle. We also discuss the effect of the hexagonal warping on the Majorana zero-energy mode and spontaneous supercurrent along the S/FI boundary.
II.
MODEL AND BASIC EQUATIONS
We consider a superconductor/ ferromagnetic insulator (S/FI) junction formed on the surface of a three dimensional topological insulator in the x-y plane, see Fig. 2(a). The S/FI boundary is perpendicular to the x-direction, so that the FI layer lies in x > 0 half-plane, while S layer in x < 0 halfplane. Using the Nambu basis the Hamiltonian for the TI surface states can be written as (we use “check” for 4 × 4 and “hat” for 2 × 2 matrices), ˆ H(k) + Mˆ ∆ˆ ˇ HS (k) = . (3) −∆ˆ −Hˆ ∗ (−k) − Mˆ Here ∆ˆ = iτˆy ∆Θ(−x), where ∆ is the superconducting pair potential, induced in the topological insulator surface due to the proximity effect, and Mˆ = σˆ z MΘ(x), where M is the spin-splitting (exchange) field of the ferromagnetic insulator, which we consider to be perpendicular to the topological surface.46,64 We use the σ notation for the Pauli matrices in spin space and τ notation for the Pauli matrices in Nambu space. The Θ(x) is the Heaviside step function. Since the superconducting correlations extend in the lateral direction on the scale of superconducting coherence length, we can consider both the superconducting pair potential and the magnetic moment in our Hamiltonian in the vicinity of the S/FI boundary at x = 0. To obtain the energy dispersion relation of the topological surface states near the S/F boundary
we start with the following equation, ˇ E − Hˇ S (k) Gˇ = 1,
(4)
where Gˇ is the Green’s function of the system at the S/FI boundary (i.e. at x = x′ = 0, where x, x′ are spatial arguments of the Green’s function), 1ˇ is the unitary 4×4 matrix, and E is the quasiparticle energy counted from the chemical potential. Introducing the crystallographic angle θ which is the azimuth angle of momentum k with respect to the x-axis, so that kx = k cos(θ ), ky = k sin(θ ), k = |k|, we can write the matrix in the left hand side of Eq. (4) in the following form, hˆ + −∆ˆ ˇ , (5) E − HS (k) ≡ ˆ ∆ hˆ − where the matrices hˆ ± are given by, E ± µ − λ k3 cos(3θ ) ∓ M ±ivke∓iθ . ∓ivke±iθ E ± µ + λ k3 cos(3θ ) ± M The e±iθ factors reflect the chiral odd-parity (p-wave) character of a topological insulator surface in proximity with an s-wave superconductor. In the previous works it was argued that its realization is a crucial requirement for a Majorana zero energy mode existence.13–17 By taking the inverse of the matrix equation Eq. (4) we can obtain the Green’s function Gˇ expressed as Gˆ Gˆ Gˇ = ˆ ee ˆ eh . (6) Ghe Ghh The diagonal blocks of the Gˇ matrix describe the propagation of the electrons and holes separately, while the off-diagonal blocks describe the interaction between the electron and hole branches, providing the mixing of the electron and hole degrees of freedom due to Andreev reflections. The electron and hole excitations in the superconductor play the role of particle and antiparticle. Electrons (filled states above the chemical potential) and holes (empty states below the chemical potential) have opposite spin and charge, but the charge difference of 2e can be absorbed as a Cooper pair in the s-wave superconducting condensate. At the chemical potential (i.e. at E = 0), the Majorana fermions are charge neutral superpositions of electrons and holes.9,10 To characterize the pair potential induced in topological surface and the Majorana zero mode we have thus to consider the the off-diagonal part of Eq. (6), i.e. the anomalous Green’s function. Since Gˆ eh and Gˆ he are related by complex conjugation it is sufficient to consider one of these matrices.
III.
SYMMETRY OF THE INDUCED PAIR POTENTIAL
Expanding Gˆ eh in Pauli matrices (where σˆ 0 is a unitary 2×2 matrix) we can write,31 (7) Gˆ eh = i f0 σˆ 0 + fx σˆ x + fy σˆ y + fz σˆ z τˆy ,
4 where f0 is the spin-singlet component (↑↓ − ↓↑), fx and fy are the combinations of equal spin triplet components, (↑↑ − ↓↓) and (↑↑ + ↓↓), correspondingly, while fz is the heterospin triplet component, (↑↓ + ↓↑).17 We can write these functions explicitly as, ∆ E 2 + M 2 − µ 2 − ∆2 − ES2 , Z+ 2∆ fx = kv [ µ sin(θ ) + iM cos(θ )] , Z+ 2∆ fy = − kv [µ cos(θ ) − iM sin(θ )] , Z+ 2∆ fz = EM − µλ k3 cos(3θ ) . Z+
f0 =
(8a) (8b) (8c) (8d)
In Eq. (8a) ES is given by the following relation, q ES = v2 k2 + λ 2 k6 cos2 (3θ ). It determines the surface band dispersion of the Fu Hamiltonian, Eq. (1), i.e. the energy dispersion relation for the bare topological insulator surface, taking into account the hexagonal warping effect,72
Symmetry of the induced pairing Even-frequency even-parity (ESE) Odd-frequency odd-parity (OSO) Even-frequency odd-parity (ETO) Odd-frequency even-parity (OTE)
spin-singlet
Energy/ spin / momentum symmetry E → −E σ ↔ σ ′ k → −k + – +
spin-singlet
–
–
–
spin-triplet
+
+
–
spin-triplet
–
+
+
TABLE I: Symmetry classification of the anomalous Green’s function. With respect to the sign change of the energy (or Matsubara frequency) and momentum, the Green function can be even (E) or odd (O). The spin part can be divided into a singlet (S) or three triplet (T) components. The pairing amplitude, given by the anomalous Green’s function, must be completely antisymmetric under the sign change of the energy, momentum, and the exchange of spin components of the electrons making up the Cooper pair. The Pauli principle allows for four different combinations; using a “energy/ spin/ momentum” notation: ESE, OSO, ETO, and OTE.
E1,2 = − µ ± ES . Here E1,2 denote the energy of upper and lower band. The function Z+ in Eqs. (8) is given by, 2 2 E+ Z± (k, E) = − 4k2 v2 (µ 2 − M 2 ) + (E 2 − M 2 )2 + E− 2 2 − E∓ (E − M)2 − E± (E + M)2 ,
(9)
where we used the following notations, q E± = v2 k2 + ∆2 + [µ ∓ λ k3 cos(3θ )]2 . Using the equalities Z± (k, E) = Z± (−k, −E), Z± (−k, E) = Z± (k, −E) = Z∓ (k, E).
(10a) (10b)
we can introduce the functions Feven = ∆/Z+ + ∆/Z− ,
Fodd = ∆/Z+ − ∆/Z−,
(11)
where Z− is defined in Eq. (9). It is easy to see that Feven is even in energy and momentum, while Fodd is odd in both arguments. We notice that changing the momentum sign k → −k is equivalent to θ → θ + π transformation. We also notice p that in case of no warping (λ = 0) one has E+ = E− = v2 k2 + ∆2 + µ 2 , so that Z+ = Z− and Fodd = 0. The function Fodd is also zero at the tips of the constant energy slowflake contour of warped Dirac cone, where cos(3θ ) = 0 [see Eq. (16) below]. In case of finite warping (λ 6= 0) but zero exchange field (no ferromagnetic insulator layer, M = 0) one also has Z+ = Z− and Fodd = 0. Now we can write explicitly the components of the anomalous Green’s function, belonging to different pairing symmetry classes. The fi functions (i = 0, x, y, z) can be written in the following symmetrized form, fi = fi+ + fi− .
(12)
For i = 0 we will have f0+ = E 2 + M 2 − µ 2 − ∆2 − ES2 Feven /2, f0− = E 2 + M 2 − µ 2 − ∆2 − ES2 Fodd /2.
(13a) (13b)
Here the f0+ belongs to the even-frequency spin-singlet evenparity (ESE) class and the f0− - to the odd-frequency spinsinglet odd-parity (OSO) class, see Table. I. We notice that for the OSO pairing realization both nonzero warping and nonzero exchange field should be present in our system. For the equal-spin triplet combinations fx,y we obtain, fx+ = kv [ µ sin(θ ) + iM cos(θ )] Feven
(14a)
= kv [ µ sin(θ ) + iM cos(θ )] Fodd ,
(14b)
= −kv [µ cos(θ ) − iM sin(θ )] Feven
(14c)
= −kv [µ cos(θ ) − iM sin(θ )] Fodd .
(14d)
fx− fy+ fy−
Here fx+ and fy+ belongs to the even-frequency spin-triplet odd-parity (ETO) class, while fx− and fy− to the odd-frequency spin-triplet even-parity (OTE) class, see Table. I. Finally the hetero-spin triplet components can be written as fz+ = EMFeven − µλ k3 cos(3θ )Fodd ,
(15a)
= EMFodd − µλ k cos(3θ )Feven .
(15b)
fz−
3
Here fz+ belongs to the OTE class, and fz− to the ETO class. We can conclude that finite warping in presence of nonzero exchange field creates a very exotic induced pair potential in the surface of a three-dimensional topological insulator in proximity with an s-wave superconductor. All allowed by the Pauli principle pairing symmetries are present in this case. We
5
a)
b)
ky
FI S
q0=p/6 kx
TI FIG. 2: (Color online) (a) Schematic illustration of the structure under consideration: superconductor/ ferromagnetic insulator (S/FI) junction formed on the surface of a three-dimensional topological insulator, TI. The x-axis is chosen perpendicular to the S/FI boundary. Two possible S/FI boundaries are shown by solid red and dashed blue lines as explained in Fig. 2(b). (b) Two possible alignments of the S/FI boundary with respect to the snowflake constant energy contour of the warped Dirac cone. The one, shown by solid red line, is favorable for the Majorana bound state realization, see details in the text. There are overall six favorable alignments at θn values, corresponding to the six tips of the snowflake contour. The favorable alignment should be perpendicular to the line, connecting the tip and the center of the snowflake contour. Arbitrary boundary alignments, for example the one, shown by dashed blue line, are unfavorable and do not host the Majorana bound states. In the latter case we predict the spontaneous supercurrent along the S/FI boundary.
note that this result is also valid for a S/TI bilayer in the presence of the perpendicular external magnetic field. Then the area where both the pair potential and the magnetic moment are present will be located at the site of a vortex.
IV.
ZERO ENERGY MAJORANA MODE
In our spin basis fx and fy are the combinations of equal spin components, (↑↑ − ↓↓) and (↑↑ + ↓↓), correspondingly, while fz is the hetero-spin component, (↑↓ + ↓↑). Therefore only fz is providing the spin-mixing, required for the realization of an electron-hole superposition due to Andreev reflections at the s-wave superconductor interface, which can form a Majorana fermion under certain conditions.9,10 For example, it was shown previously that odd in energy triplet fz component is required for the Majorana zero mode existence in a one-dimensional nano-wire.13 Let us assume that same is valid in case of the 2D surface states of a three dimensional topological insulator.15 Then the hetero-spin OTE pairing appears naturally in isotropic models with no warping (λ = 0) when finite out-of-plane magnetic field is applied to the structure (or the exchange field in our consideration). In this case Fodd = 0 and the sole odd-frequency component is fz = EMFeven , which belongs to the OTE symmetry class15,17 and appears only when three ingredients 1) the topological surface states, 2) superconducting pair potential ∆ due to the proximity effect, and 3) nonzero exchange field M in the zaxis direction, are brought together. When any of these ingredients is missing the anomalous Green’s function loses the odd-frequency pairing symmetry and the Majorana fermion is absent.13–17 From Eqs. (15) we clearly see that at finite warping (λ 6= 0) the fz component is no more odd in energy, since the fz+ term is even in energy for any chosen k direction. The z-component
of the anomalous Green’s function became odd in energy only at the following six values of the crystallographic angle θ ,
θn = π /6 + π n/3,
(16)
which correspond to the six tips of the snowflake constant energy contour, shown in Fig. 1 (n is an integer number). At θ 6= θn the fz component is neither odd nor even in energy. We note that non-zero out-of-plane exchange field is crucial for the Majorana mode realization, since at M = 0 the oddfrequency pairing component fz− disappears even in case of finite warping. Let us now consider six lines θ = θn on the TI surface, where the topological surface spectrum is given by the following relation, r q ± (∓)
µ 2 + ∆2 + v2 k2 + M 2 ± 2 µ 2 v2 k2 + M 2 [µ 2 + ∆2 ].
In this expression we need to take all possible combinations of ± signs to get four energy bands. In the limit of µ ≫ ∆, which is often the case, this can in good approximation be written as16 p ± (∓)µ ± v2 k2 + M 2 .
For now, we focus on the hybrid structure in Fig. 2(a), which was previously considered in Ref. 46. In order to have a zero-energy Majorana mode for θ = θn , it needs to be localized both at the superconducting side and at the side of the ferromagnetic insulator. At the superconducting side it happens due to the superconducting gap. At the FI side the exchange field M has to be large enough p so that the chemical potential is inside the gap,15 M > ∆2 + µ 2, or just M > µ if µ ≫ ∆. Then the zero-energy mode is fully localized. As was shown in Ref. 15 the midgap Andreev bound state became the Majorana zero mode only when incident electron
6 and the Andreev-reflected hole trajectories are perpendicular to the S/FI boundary. As follows from the above arguments to insure the MF existence the S/FI boundary should be aligned perpendicular to the line θ = θn . In Fig. 2(b) we show the alignment of the S/FI boundary (shown by solid red line) which insures the Majorana zero energy mode existence in particular case θ0 = π /6. Generally, the boundary should be perpendicular to the line θ = θn in the topological surface plane and totally six favorable alignments are possible. The expressions for the Majorana zero modes given in Ref. 15 hold in our case for θ = θn . In the vicinity of θ = θn the bound state energy is E = −∆ sin(θ − θn ) and goes to zero at θ = θn . For θ 6= θn , the surface bound states dispersion relations for finite λ become very cumbersome and we do not present them here. But, importantly, based on our assumption of direct correspondence between the hetero-spin OTE component and the Majorana fermion, one can provide the following general symmetry-based argument. Since the triplet component fz of the anomalous Green’s function, responsible for the MF occurrence, is neither odd nor even in energy in this case [see Eqs. (15)], no zero-energy Majorana bound states will be formed for any alignment of the S/FI boundary except six aforementioned. The example of such unfavorable alignment is shown in Fig. 2(b) by a blue dashed line. This provides an important selection rule to the realization of Majorana modes in S/FI hybrid structures, formed on the topological insulator surface. It also gives a guideline for the observation of the Majorana fermion in experiments. Recently another type of selection rules for the MF realization were established in Ref. 80. The situation when the crystallographic angle θ is in the intermediate position between tips of the snowflake structure, θ 6= θn is rather complicated and will be considered numerically elsewhere. We can make some predictions in this case using the general symmetry arguments. We will again employ the analogy with a one dimensional topological nano-wire case, where analytical solution is easy to obtain. The Hamiltonian of a three dimensional topological insulator warped surface state, Eqs. (1)-(2) in the presence of a magnetic field (or an exchange field M of the ferromagnetic insulator), perpendicular to the surface can be written as, Hˆ M (k) = −µ + v(kx σˆ y − ky σˆ x )+ σˆ z λ k3 cos(3θ )+ σˆ z M. (17) We consider the S/FI boundary perpendicular to the line θ = θ ′ , where θ ′ 6= θn . Let us project this Hamiltonian on the direction along the S/FI boundary, i.e. on the y-axis (the S/FI boundary is located at x = 0). Then the effective onedimensional Hamiltonian for the Andreev bound states will look like (kx ∼ 0), Hˆ eff (ky ) = − µ − vky σˆ x + σˆ z λ ky3 cos(3θ ) + σˆ zM.
(18)
From the viewpoint of the time reversal and spatial symmetries it is equivalent to the following one-dimensional Hamiltonian of a topological nano-wire, which was recently considered in Ref. 81, Hˆ wire (ky ) = −µ + α ky σˆ y + Mσˆ ,
(19)
where Mσˆ = Mx σˆ x + My σˆ y + Mz σˆ z , and α is some constant (Mi 6= 0, i = x, y, z). It was shown in Ref. 81 that a Josephson junction through such nano-wire will be in the ϕ -state.82 In our model it will correspond to the spontaneous supercurrent along the S/FI boundary in Fig. 2.83 Therefore, from the above symmetry arguments and using the assumption about the equivalence of the MF and heterospin OTE pairing, we can end up with a following picture. When θ = θn in Eq. (16), there is a possibility of the Majorana zero mode realization if the S/FI boundary is perpendicular to the θ = θn direction (see Fig. 2). In the same time the spontaneous supercurrent state is absent. But while we change the boundary alignment so that it becomes perpendicular to the θ = θ ′ direction, where θ ′ 6= θn , the warping term is “switched on” and we predict the spontaneous current along the S/FI boundary, and no zero energy Majorana mode.
V.
CONCLUSION
In conclusion, we have theoretically discussed the proximity effect in three-dimensional topological insulators with warped surface state, in the presence of magnetic moment, perpendicular to the TI surface. For this we have considered the superconductor/ ferromagnetic insulator heterostructure, formed on the surface of the topological insulator, see Fig. 2. This theory is also applicable to a S/TI bilayer in the presence of the external magnetic field. Then the area where both the pair potential and the magnetic moment are present will be located at the site of a vortex. We have extensively discussed the symmetry of the superconducting pair potential, induced in the warped TI surface state by proximity with a superconductor. The potential is very exotic with all possible components, allowed by the Pauli principle. Finally, we made some predictions about the Majorana zero mode, known to exists in the structure under consideration. If we employ the assumption that the Majorana fermion and the odd-frequency spin-triplet evenparity hetero-spin component are directly related to each other (which should be the case in the one-dimensional nano-wire, see Refs. 13–15), then the Majorana fermion realization is sensible to the orientation of the S/FI boundary with respect to the snowflake constant energy contour of the warped Dirac cone. The favorable alignment should be perpendicular to the line, connecting the snowflake tip and the center of the snowflake contour. Arbitrary boundary alignments are unfavorable and do not host the Majorana bound states, but in this case we predict the spontaneous supercurrent flow along the S/FI boundary.
Acknowledgments
The authors thank F.S. Bergeret, P. Burset, J. Linder, and especially I.V. Tokatly for useful discussions, and A.A. Zarudneva for the help in the manuscript preparation. The research carried out by A.S. Vasenko was supported within the framework of the Academic Fund Program at the National
7 Research University Higher School of Economics (HSE) in 2016 - 2017 (grant No. 16-01-0051) and supported within the framework of a subsidy granted to the HSE by the Government of the Russian Federation for the implementation of the Global Competitiveness Program. The research carried out by
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